Embedded newform 1764.2.a.e.1.1

• Label: 1764.2.a.e.1.1
• Level: $1764$
• Weight: $2$
• Character: 1764.1
• Self dual: yes
• Analytic conductor: $14.086$
• Analytic rank: $1$
• Dimension: $1$
• CM discriminant: -3
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1764.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(14.0856109166\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 36)
Fricke sign:	\(+1\)
Sato-Tate group:	$N(\mathrm{U}(1))$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1764.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-2.00000 q^{13} -8.00000 q^{19} -5.00000 q^{25} +4.00000 q^{31} -10.0000 q^{37} +8.00000 q^{43} -14.0000 q^{61} -16.0000 q^{67} +10.0000 q^{73} -4.00000 q^{79} -14.0000 q^{97} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0
\(5\)	0	0					−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(7\)	0	0
			\(11\)	0	0		−	1.00000i	\(-0.5\pi\)
		1.00000i				\(0.5\pi\)
\(13\)	−2.00000	−0.554700	−0.277350	−	0.960769i	\(-0.589456\pi\)
			−0.277350	+	0.960769i	\(0.589456\pi\)
\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(19\)	−8.00000	−1.83533	−0.917663	−	0.397360i	\(-0.869927\pi\)
			−0.917663	+	0.397360i	\(0.869927\pi\)
\(23\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(29\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(31\)	4.00000	0.718421	0.359211	−	0.933257i	\(-0.383046\pi\)
			0.359211	+	0.933257i	\(0.383046\pi\)
\(37\)	−10.0000	−1.64399	−0.821995	−	0.569495i	\(-0.807139\pi\)
			−0.821995	+	0.569495i	\(0.807139\pi\)
\(41\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(43\)	8.00000	1.21999	0.609994	−	0.792406i	\(-0.291172\pi\)
			0.609994	+	0.792406i	\(0.291172\pi\)
\(47\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1764.2.a.e.1.1		1
3.2	odd	2	CM	1764.2.a.e.1.1		1
4.3	odd	2		7056.2.a.bb.1.1		1
7.2	even	3		1764.2.k.g.361.1		2
7.3	odd	6		1764.2.k.h.1549.1		2
7.4	even	3		1764.2.k.g.1549.1		2
7.5	odd	6		1764.2.k.h.361.1		2
7.6	odd	2		36.2.a.a.1.1	✓	1
12.11	even	2		7056.2.a.bb.1.1		1
21.2	odd	6		1764.2.k.g.361.1		2
21.5	even	6		1764.2.k.h.361.1		2
21.11	odd	6		1764.2.k.g.1549.1		2
21.17	even	6		1764.2.k.h.1549.1		2
21.20	even	2		36.2.a.a.1.1	✓	1
28.27	even	2		144.2.a.a.1.1		1
35.13	even	4		900.2.d.b.649.2		2
35.27	even	4		900.2.d.b.649.1		2
35.34	odd	2		900.2.a.g.1.1		1
56.13	odd	2		576.2.a.e.1.1		1
56.27	even	2		576.2.a.f.1.1		1
63.13	odd	6		324.2.e.c.217.1		2
63.20	even	6		324.2.e.c.109.1		2
63.34	odd	6		324.2.e.c.109.1		2
63.41	even	6		324.2.e.c.217.1		2
77.76	even	2		4356.2.a.g.1.1		1
84.83	odd	2		144.2.a.a.1.1		1
91.34	even	4		6084.2.b.f.4393.2		2
91.83	even	4		6084.2.b.f.4393.1		2
91.90	odd	2		6084.2.a.i.1.1		1
105.62	odd	4		900.2.d.b.649.1		2
105.83	odd	4		900.2.d.b.649.2		2
105.104	even	2		900.2.a.g.1.1		1
112.13	odd	4		2304.2.d.q.1153.2		2
112.27	even	4		2304.2.d.a.1153.1		2
112.69	odd	4		2304.2.d.q.1153.1		2
112.83	even	4		2304.2.d.a.1153.2		2
140.27	odd	4		3600.2.f.m.2449.2		2
140.83	odd	4		3600.2.f.m.2449.1		2
140.139	even	2		3600.2.a.e.1.1		1
168.83	odd	2		576.2.a.f.1.1		1
168.125	even	2		576.2.a.e.1.1		1
231.230	odd	2		4356.2.a.g.1.1		1
252.83	odd	6		1296.2.i.h.433.1		2
252.139	even	6		1296.2.i.h.865.1		2
252.167	odd	6		1296.2.i.h.865.1		2
252.223	even	6		1296.2.i.h.433.1		2
273.83	odd	4		6084.2.b.f.4393.1		2
273.125	odd	4		6084.2.b.f.4393.2		2
273.272	even	2		6084.2.a.i.1.1		1
336.83	odd	4		2304.2.d.a.1153.2		2
336.125	even	4		2304.2.d.q.1153.2		2
336.251	odd	4		2304.2.d.a.1153.1		2
336.293	even	4		2304.2.d.q.1153.1		2
420.83	even	4		3600.2.f.m.2449.1		2
420.167	even	4		3600.2.f.m.2449.2		2
420.419	odd	2		3600.2.a.e.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.2.a.a.1.1	✓	1	7.6	odd	2
36.2.a.a.1.1	✓	1	21.20	even	2
144.2.a.a.1.1		1	28.27	even	2
144.2.a.a.1.1		1	84.83	odd	2
324.2.e.c.109.1		2	63.20	even	6
324.2.e.c.109.1		2	63.34	odd	6
324.2.e.c.217.1		2	63.13	odd	6
324.2.e.c.217.1		2	63.41	even	6
576.2.a.e.1.1		1	56.13	odd	2
576.2.a.e.1.1		1	168.125	even	2
576.2.a.f.1.1		1	56.27	even	2
576.2.a.f.1.1		1	168.83	odd	2
900.2.a.g.1.1		1	35.34	odd	2
900.2.a.g.1.1		1	105.104	even	2
900.2.d.b.649.1		2	35.27	even	4
900.2.d.b.649.1		2	105.62	odd	4
900.2.d.b.649.2		2	35.13	even	4
900.2.d.b.649.2		2	105.83	odd	4
1296.2.i.h.433.1		2	252.83	odd	6
1296.2.i.h.433.1		2	252.223	even	6
1296.2.i.h.865.1		2	252.139	even	6
1296.2.i.h.865.1		2	252.167	odd	6
1764.2.a.e.1.1		1	1.1	even	1	trivial
1764.2.a.e.1.1		1	3.2	odd	2	CM
1764.2.k.g.361.1		2	7.2	even	3
1764.2.k.g.361.1		2	21.2	odd	6
1764.2.k.g.1549.1		2	7.4	even	3
1764.2.k.g.1549.1		2	21.11	odd	6
1764.2.k.h.361.1		2	7.5	odd	6
1764.2.k.h.361.1		2	21.5	even	6
1764.2.k.h.1549.1		2	7.3	odd	6
1764.2.k.h.1549.1		2	21.17	even	6
2304.2.d.a.1153.1		2	112.27	even	4
2304.2.d.a.1153.1		2	336.251	odd	4
2304.2.d.a.1153.2		2	112.83	even	4
2304.2.d.a.1153.2		2	336.83	odd	4
2304.2.d.q.1153.1		2	112.69	odd	4
2304.2.d.q.1153.1		2	336.293	even	4
2304.2.d.q.1153.2		2	112.13	odd	4
2304.2.d.q.1153.2		2	336.125	even	4
3600.2.a.e.1.1		1	140.139	even	2
3600.2.a.e.1.1		1	420.419	odd	2
3600.2.f.m.2449.1		2	140.83	odd	4
3600.2.f.m.2449.1		2	420.83	even	4
3600.2.f.m.2449.2		2	140.27	odd	4
3600.2.f.m.2449.2		2	420.167	even	4
4356.2.a.g.1.1		1	77.76	even	2
4356.2.a.g.1.1		1	231.230	odd	2
6084.2.a.i.1.1		1	91.90	odd	2
6084.2.a.i.1.1		1	273.272	even	2
6084.2.b.f.4393.1		2	91.83	even	4
6084.2.b.f.4393.1		2	273.83	odd	4
6084.2.b.f.4393.2		2	91.34	even	4
6084.2.b.f.4393.2		2	273.125	odd	4
7056.2.a.bb.1.1		1	4.3	odd	2
7056.2.a.bb.1.1		1	12.11	even	2